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(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 6, June 2011
Additive model of reliability of biometric systems
with exponential distribution of failure probability
Bihać, Bosnia and Hercegovina
Zoran Ćosić(Author) jacosic@gmail.com
director
Statheros d.o.o.
Kaštel Stari, Croatia Miroslav Bača (Author)
zoran.cosic@statheros.hr professor
Faculty of Organisational and Informational science
Varaždin, Croatia
Jasmin Ćosić (Author) miroslav.baca@foi.hr
IT Section of Police Administration
Ministry of Interior of Una-sana canton
Abstract— Approaches for reliability analysis of biometric Data collection subsystem consists of a biometric sample,
systems are subject to a review of numerous scientific papers. method of sampling, and sensors that are sampled. Signal
Most of them consider issues of reliability of component software processing subsystem consists of drainage structures, quality
applications. System reliability, considering technical and control and comparison of samples. Decision subsystem
software part, is of crucial importance for users and for
manufacturers of biometric systems.
consists of the decision mechanisms and storage subsystems.
In this paper, the authors developed a mathematical model to Schematisation of model described in figure 1 can be shown
analyse the reliability of biometric systems, regarding the on figure 2.
dependence of components with exponential distribution of
failure probability.
Keywords- Additive model, Biometric system, reliability,
exponential distribution, UML,
Figure 2
I. INTRODUCTION
Schematic presentation of a biometric [1] system is a
The general biometric system, according to [1] Wyman shown simplified representation of a system in Figure 1 and shows
in Figure 1, consists of 5 elements which are located in all the serial configuration of system components dependence
biometric systems today.
II. THE DEFINITION OF THE RELIABILITY OF BIOMETRIC
SYSTEMS
Biometric system designers and producers are motivated to
use already constructed components and modules. Component
system has a high reliability expectations , no matter who is
producer. Most of existing reliability models are so generally
described to not consider particularity of components and
modules. In this paper authors will describe methodology
based on mathematical model which take account of
component reliability and connections reliability also. UML
methodology in describing system and his inner interaction,
simplify approach for researchers.
Figure 1. [1] UML [2] is also becoming standard in the process of designing
and manufacturing systems so production of component
Each subsystem consists of the elements that contribute to the systems gets benefits from the UML representation.
overall system quality.
1 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 6, June 2011
Assessment of [3] generic biometric system reliability in UML
is given in Figure 3, which describes the use of Use Case Diagrams sequences play an important role in assessing the
diagram: reliability of the system because they give information on how
many components are involved in the execution of a scenario.
Through sequence diagrams it is simple to count the periods of
availability of components in the given scenarios as shown in
figure (3). The probability of failure of components with
known busy periods, can be given by the following
expression:
(3)
Figure 3 [3]
In witch is:
- - probability of component failure i in the scenario j
q1 and q2 represent the probability that users u1 and u2 will - - occupancy time of component i in the scenario j
access the system using some of its functionality.
P11 and P12 represent the probability that user u1 will use the The expression (3) applies only if the following conditions are
functionality of f1 and f2, and P21 and P22 represent the same met:
probability for the user u2.
- Independence of failure: the probability of failure of
The probability of execution of use case x, is defined by the
one component does not depend on other components
expression:
- Regularity of failure: the probability of failure of one
component is equal throughout the execution of
(1) occupation period of the component
m is number of users. You can also show every moment of occupancy of any
component of the system considering the method to be
If we are able to join a no uniform distribution to Diagram of executed at that moment in the scenario
sequences in a given use case then (1) can be expressed as:
If we replace with a set of method of failure probability
(2) where is then equation (3) becomes:
(4)
Where the fj (k) - frequency of the k-th transition of sequence
diagram in the j-th case. P (kj) – presents a probability of
default scenarios. During system operation, components interact and exchange
information. Then it is necessary to take into consideration
occupation period of the component:
(5)
Where is θij- the probability of failure of system components
and bp-busy period of the system.
(6)
Where is Ψlmj- the probability of failure connections between
the components and - the number of
interactions between system components.
(7)
Figure 4 Sequence diagram [3] The reliability of the system taking into account the
probability of failure can be expressed as:
2 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 6, June 2011
n2( Δt ) - number of failures in certain time interval Δt
n1( t − Δt ) – the non failed number of elements at the end of
(7a)
the interval Δt , or until t − Δt
The intensity of the element failure is calculated by the
(7b) expression:
1
) 1
III. RELIABILITY PREDICTION OF THE COMPONENTS SERIAL λEL =Θ= (11)
DEPENDENCE WITH EXPONENTIAL DISTRIBUTION n Θ⋅n
Based on the above assumptions [4], [5] system reliability can Where is:
be calculated according to the law of exponential distribution:
n- number of usable parts of the confidence interval
(1 − α ) = 0, 75
(8)
Θ - lower limit of confidence for the mean time between
In wich are: failures
Rs - System reliability
λ - Intensity of system fault The total intensity of failure taking into consideration the
t - Required time of reliable operation of system number of elements that are not failed in a given time is
calculated by formula:
Proof:
(12)
Where is:
nEL- number of elements of subsystems that are not failed
Mean time between failures MTBFs can be calculated using
In a serial dependence between of all parts of the system the the expression:
failure of any part of the system may cause the failure of the
entire system.
Failure intensity function λs in the case of a serial dependence
of system elements is calculated by the expression: (13)
(9)
Calculation of reliability [6] of some element is based on
Where is λi - failure intensity of the i-th part of the system.
empirical data on the time of functioning and eventual failure
of the element.
Failure intensity function λs is equal to the ratio between the
Problem [7], [8] becomes more complex when the information
number of failures in the time-frame and the correct number of
about the failure doesn't exist. In case that part worked
elements in the system, until the beginning of this interval:
perfectly, and information about the exploitation are available.
If one assumes that the given part can apply the rule of the
exponential distribution, it is possible to determine the upper
(10) limit of confidence for the intensity of failure, in the cases of
continuous operation or one or more failures.
Where is:
λs -function of failure intensity of system Lower limit of confidence for the mean time between
)
Δt - failure time of an system element failures Θ , for confidence interval (1 − α ) is calculated using
the formula:
3 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 6, June 2011
probability. In accordance with this authors will define
) 2tr 2tr mathematical model for recovery system probability than
Θ≥ = (14) system readiness to use.
χ 2α ,2 r + 2 χ 20.25,2
Where is:
REFERENCES
tr – total time of system operation
r – Number of elements that have failed [1] Zasnivanje otvorene ontologije odabranih segmenata biometrijske
znanosti - Markus Schatten– Magistarski rad – FOI 2007
χ 2α ,2r + 2 - Random variable which has distribution [2] Modelling biometric systems in UML – Miroslav Bača, Markus
Schatten, Bernardo Golenja, JIOS 2007 FOI Varaždin
[3] A Bayesian Approach to Reliability Prediction and Assessment of
Component
IV. SPECIAL CASE OF NOT-FAILURE SYSTEM [4] Based Systems – KH. Singhy, V. Cortellessa, B. Cukic, E.
Gunely,V.Bharadwaj
Considering (8), (10) and (14) we obtain an expression for the
[5] Department of Statistics, Lane Department of Computer Science and
reliability of the whole system: Electrical Engineering West Virginia University, Proceedings of the
12th International Symposium on Software Reliability Engineering
(15) (ISSREí01) 1071-9458/01 2001 IEEE
[6] Simulacijsko modeliranje pouzdanosti tehničkog sustava brodskog
In futher calculations: kompresora – Zoran Ćosić – Magistarski rad - 2007
[7] Teorija pouzdanosti tehničkih sistema, Vujanović Nikola,
Vojnoizdavački novinski centar, Beograd 2005
(16) [8] The Impact of Error Propagation on Software Reliability Analysis of
Component-based Systems – Petar Popic, Master thesis, West Vriginia
2005
Taking into account (11) we obtain an expression for the
reliability of the each elements of the system:
AUTHORS PROFILE
Zoran Ćosić, CEO at Statheros ltd, and business consultant in business process
standardization field. He received BEng degree at Faculty of nautical
(17) science , Split (HR) in 1990, MSc degree at Faculty of nautical science ,
Split (HR) in 2007 , actually he is a PhD candidate at Faculty of
informational and Organisational science Varaždin Croatia. He is
a member of various professional societies and program
V. CONCLUSION AND FURTHER RESEARCH committee members. He is author or co-
author more than 20 scientific and professional papers. His main
The reliability of technical systems is the subject for many fields of interest are: Informational security, biometrics and privacy,
research scientists, according to that analysis are available in business process reingeenering,
different models of reliability of biometric systems that in Jasmin Ćosić has received his BE (Economics) degree from University of
most cases take into account the software as one of the Bihać, B&H in 1997. He completed his study in Information Technology
field (dipl.ing.Information Technlogy) in Mostar, University of Džemal
components of the same system. This study developed Bijedić, B&H. Currently he is PhD candidate in Faculty of Organization
mathematical model of a generic biometric system that and Informatics in Varaždin, University of Zagreb, Croatia. He is
consider user, hardware and software influence and assumes working in Ministry of the Interior of Una-sana canton, B&H. He is a
serial dependence of the components and the exponential ICT Expert Witness, and is a member of Association of Informatics of
B&H, Member of IEEE and ACM. His areas of interests are Digital
distribution of failure probability of system components. Forensic, Computer Crime, Information Security and DBM Systems. He
Scientific contribution is expressed through mathematical has presented and published over 20 conference proceedings and journal
model definition of biometric system reliability prediction articles in his research area
within special case where is not possible to get failure data or Miroslav Bača is currently an Associate professor, University of Zagreb,
in the case of perfectly working system. The results of Faculty of Organization and Informatics. He is
a member of various professional societies and program
calculations can prevent real failure events by defining committee members, and he is reviewer of several international
preventive maintenance. journals and conferences. He is also the head of the Biometrics centre in
This mathematical model allows the prediction of system Varaždin, Croatia. He is author or co-
author more than 70 scientific and professional papers and two
reliability in the early fase of projecting of its components . books. His main research fields are computer forensics, biometrics and
The subject of further research will be to create an privacy professor at Faculty of informational and Organisational science
integrated mathematical model of reliability of complex Varaždin Croatia
systems that considers the parallel and combined dependency
of system components with different distributions of failure
4 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
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